Part II Data and Methods
14.4 DJI Family Surveys
1.5
C55 C60
C65
Fig. 14.3-3 Cumulated cohort birth rates calculated from Table 14.3-2 for three cohorts in the western part of Germany (solid lines). The dotted lines show corresponding rates calculated from the SOEP (C55) and the GLHS (C60).
1970 1980 1990 2000
0 0.5 1 1.5
West East C55
C60 C65
Fig. 14.3-3 Cumulated cohort birth rates calculated from the data in Table 14.3-2 for the western part (solid lines) and the eastern part (dotted lines) of Germany.
amounts to the number of children without a valid birth year. However, the impact of these missing values oncumulated cohort birth rates is quite limited, and so the data can nevertheless be used for further investigation.
Figure 14.3-3 shows these cumulated rates for the cohorts in the western part of Germany and, for cohorts C55 and C60, also provides a comparison with the results from theSOEP and the GLHSdata, respectively. Figure 14.3-4 compares the rates between both territories.
14.4 DJI Family Surveys
1.A further source of information about childbearing histories in Germany is a series of surveys conducted by the Deutsches Familieninstitut (DJI, M¨unchen). Data sets are available from the Zentralarchiv f¨ur empirische Sozialforschung (K¨oln). In the present section we use data from a survey conducted in the territory of the formerFRGin 1988. The sample refers to persons with a German citizenship who, at the interview date in 1988, lived in private households and were between 18 and 55 years old.19 The final sample size is 10043, 4554 men and 5489 women. The following table shows the distribution of birth years of the female participants:
Birth year Number Birth year Number Birth year Number
1933 130 1946 96 1959 161
1934 121 1947 128 1960 169
1935 129 1948 165 1961 158
1936 153 1949 143 1962 158
1937 143 1950 158 1963 165
1938 139 1951 173 1964 179
1939 149 1952 172 1965 145
1940 142 1953 153 1966 147
1941 123 1954 166 1967 112
1942 129 1955 185 1968 118
1943 148 1956 157 1969 114
1944 137 1957 182 1970 66
1945 96 1958 180
For compatibility with the data discussed in previous sections we consider the following birth cohorts:
Birth cohort Birth years Number of women
C35 1933−1937 676
C40 1938−1942 682
C45 1943−1947 605
C50 1948−1952 811
C55 1953−1957 843
C60 1958−1962 826
Women born later than 1962 will not be considered because their age in 1988 does not allow any reliable conclusions about childbearing histories.
2.As was done in the previous sections, we begin with an investigation of ages at first childbearing. This is easy because the data set already contains
19For a description of the sampling design see Alt (1991).
Table 14.4-1 Age at first childbearing in our DJI subsample.
C35 C40 C45 C50 C55 C60
τ d= 1 d= 0 d= 1 d= 0 d= 1 d= 0 d= 1 d= 0 d= 1 d= 0 d= 1 d= 0
Fig. 14.4-1 Comparison of survivor functions for the age at first child-bearing for birth cohorts C35, C40, C45, C50, C55, and C60.
a variable providing the age of women at first childbearing.20 Table 14.4-1 shows, separately for birth cohorts, how many women of specified age have given birth to a child (d= 1) or are censored at the interview date (d = 0).21 These data can be used to estimate survivor functions as in the previous sections. Figure 14.4-1 compares the survivor functions with estimates based on theGLHS SOEP andFFSdata. For birth cohorts C35, C40, C45, and C50, the results are quite similar. Substantial differences only occur for the two younger cohorts, C55 and C60.
20We have used the SPSS filefall88.sav. The variable providing age at first child-bearing isF275 ALT.
21Notice that for some birth cohorts the totals are slightly smaller than the number of cases tabulated in the preceeding paragraph because we have dropped cases with a reported age at first childbearing below 15.
Birth Rates in East Germany
This chapter is not finished yet.
In- and Out-Migration
This chapter is not finished yet.
Chapter 17
An Analytical Modeling Approach
In the present chapter we begin with the discussion of an analytical model that can support modal reasoning about demographic processes. We begin with a version of the model that takes into account births and deaths but ignores migration. How to extend the model in order to include migration will be discussed in Section 18.6.
17.1 Conceptual Framework
1.To introduce a conceptual framework for the model, we refer to a de-mographic process, (S,T∗,Ωt), as discussed in Section 3.2. S provides the spatial context,T∗ is the time axis, and Ωt represents the population living in the spaceSin the temporal locationt∈T∗. The numbers of men and women in Ωt agedτ will be denoted bynmt,τ andnft,τ, respectively; the total number of persons agedτ will be denoted bynt,τ:=nmt,τ+nft,τ. To simplify notations we will assume that age is measured in the same time units that are used in the definition ofT∗. For example, if T∗ refers to calendar years, it will be assumed that age is measured in completed years.
We also assume a maximal age which will be denoted byτm.1
2.To formulate the model it is now helpful to use matrix notations.2 Clas-sified by age, the male and female population will be represented, respec-tively, by the vectors Notice that the count of vector elements begins with 1, not with 0, so that only persons who have reached an age of one time unit will be given an explicit representation.
3.The purpose of a demographic model is to provide a conceptual frame-work for thinking about possible developments of a population:
n0 −→ n1 −→ n2 −→ · · ·
1This is not a serious limitation because τmcan be given an arbitrarily high value;
also, in practical applications,τmcan be assumed to be an open-ended age class.
2For a brief introduction to matrix notations and elementary rules see Rohwer and P¨otter (2002a, Appendix A).
that begin in some arbitrary temporal location with an initial population Ω0, here represented by the vector n0. This requires the introduction of rules that can be used to deriven1fromn0,n2fromn1, and so on. Since we ignore migration (think ofS as a closed region), it suffices to take into account birth and death events. However, only women can give birth to children, and so it is necessary to represent the process in the following way:
nm0 −→ nm1 −→ nm2 −→ · · ·
% % %
nf0 −→ nf1 −→ nf2 −→ · · ·
4.In order to formulate rules we use age-specific birth and death rates.
Death rates for men and women at age τ in temporal location t will be denoted, respectively, by
δt,τm and δt,τf
Given these rates, the number of men and women dying in t at age τ is δmt,τnmt,τ and δft,τnft,τ, respectively. Notice that the assumption of a maximal ageτmimplies thatδmt,τm=δft,τm = 1.
5.Age-specific birth rates will be denoted by βt,τ∗ .3 In order to simplify the formulation of the model these rates will be interpreted as follows:
βt,τ∗ nft,τ is the number of children, born of women at age τ in temporal locationt, who survived the first time unit and are consequently members of Ωt+1. Of course, since only women can bear children, these birth rates need not be indexed with respect to sex. However, one has to take into account differences in the percentages of male and female births. We use σm and σf to denote the proportions (σm+σf = 1). Therefore, if nt+1,1
is the total number of children born int, the number of male children is nmt+1,1=σmnt+1,1and the number of female children isnft+1,1=σfnt+1,1. To ease notations, we assume that the sex ratio at birth is independent of mother’s age and constant over time.
6.Since we only consider children who survived the first time unit we also do not explicitly model death rates of children during the temporal location in which they are born. There is, however, a simple relationship betweenβ∗t,τ and the birth ratesβt,τ, introduced in Section 11.1:
βt,τ∗ =βt,τ(1−δt,0)
In this formulation,δt,0=σmδt,0m +σfδft,0is a weighted mean of the death rates of male and female children during their first year of life.
3We assume that these birth rates are defined for all ages and have a value of zero at ages outside the reproductive period of women.
7.Assuming that birth and death rates are given, one can derive some elementary rules for the development of the population. First, the total number of children born in temporal locationtand still alive int+ 1 can be derived fromnft and the age-specific birth rates as follows:
nt+1,1 =
τm
X
τ=1
βt,τ∗ nft,τ
Secondly, the relation between the number of men and women at ages τ ≥ 1 in two successive temporal locations can be derived from death rates:
nmt+1,τ+1 = (1−δt,τm)nmt,τ and nft+1,τ+1 = (1−δt,τf )nft,τ
Together, the three equations allow to derivenmt andnft fromnm0 andnf0 for allt >0. Of course, this requires to think of the birth and death rates, and also the proportions of male and female births, as given and known parameters of the demographic process.
8.We now proceed with matrix notation. First, we define (τm, τm) matri-ces
which comprise the age-specific birth rates. The number of male and female children int+ 1 is then given, respectively, by
death rates of men and women:
Dm,t := the three equations derived in the previous paragraph can be written as
nmt+1 = Dm,tnmt +σmBtnft nft+1 = Df,tnft +σfBtnft
(17.1.2)